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NASA NTRS1993

Probability density distribution of velocity differences at high Reynolds numbers

Recent understanding of fine-scale turbulence structure in high Reynolds number flows is mostly based on Kolmogorov's original and revised models. The main finding of these models is that intrinsic characteristics of fine-scale fluctuations are universal ones at high Reynolds numbers, i.e., the functional behavior of any small-scale parameter is the same in all flows if the Reynolds number is high enough. The only large-scale quantity that directly affects small-scale fluctuations is the energy flux through a cascade. In dynamical equilibrium between large- and small-scale motions, this flux is equal to the mean rate of energy dissipation epsilon. The pdd of velocity difference is a very important characteristic for both the basic understanding of fully developed turbulence and engineering problems. Hence, it is important to test the findings: (1) the functional behavior of the tails of the probability density distribution (pdd) represented by P(delta(u)) is proportional to exp(-b(r) absolute value of delta(u)/sigma(sub delta(u))) and (2) the logarithmic decrement b(r) scales as b(r) is proportional to r(sup 0.15) when separation r lies in the inertial subrange in high Reynolds number laboratory shear flows.

Praskovsky, Alexander A.↗

NASA NTRSDate not supplied

Bias in Planning Algorithms

Does bias exist in planning algorithms? If so, how does bias manifest, and how important is this bias? Answering this question requires a formal, mathematical definition of bias. We formally define bias as the distance between the probability distributions of solutions returned by various algorithms, and the uniform distribution over solutions. We show in this paper that deterministic algorithms are inherently biased, as they don’t return all solutions, and that this property holds even when algorithms return a set of plans instead of just one plan. Exceptions are problem instances or problem classes for which only a single solution exists. We then discuss changing the definition of bias to compare the probability distributions of properties of sets of plans instead of individual plans. We show the property bias is smaller than the bias of actual plans. Finally, we show that entropy is a proxy for the more complex and more expensive distance measurement between pairs of probability distributions. We then describe a roadmap for future investigations of bias in planning.

Planning Scheduling Algorithms↗

NASA NTRS1987

Comparing hard and soft prior bounds in geophysical inverse problems

In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about X sub E is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. Two forms of prior information are compared: a soft bound on X sub E is a probability distribution p sub x on X which describeds the observer's opinion about where X sub E is likely to be in X; a hard bound on X sub E is an inequality Q sub x(X sub E, X sub E) is equal to or less than 1, where Q sub x is a positive definite quadratic form on X. A hard bound Q sub x can be softened to many different probability distributions p sub x, but all these p sub x's carry much new information about X sub E which is absent from Q sub x, and some information which contradicts Q sub x. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p sub x. If that probability distribution was obtained by softening a hard prior bound Q sub x, rather than by objective statistical inference independent of y, then p sub x contains so much unsupported new information absent from Q sub x that conclusions about z obtained with SI or BI would seen to be suspect.

Backus, George E.↗

NASA NTRS1988

Comparing hard and soft prior bounds in geophysical inverse problems

In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about X sub E is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. Two forms of prior information are compared: a soft bound on X sub E is a probability distribution p sub x on X which describes the observer's opinion about where X sub E is likely to be in X; a hard bound on X sub E is an inequality Q sub x(X sub E, X sub E) is equal to or less than 1, where Q sub x is a positive definite quadratic form on X. A hard bound Q sub x can be softened to many different probability distributions p sub x, but all these p sub x's carry much new information about X sub E which is absent from Q sub x, and some information which contradicts Q sub x. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p sub x. If that probability distribution was obtained by softening a hard prior bound Q sub x, rather than by objective statistical inference independent of y, then p sub x contains so much unsupported new information absent from Q sub x that conclusions about z obtained with SI or BI would seen to be suspect.

Backus, George E.↗

DOE OSTI2024

Uncertainty Propagation within Chained Models for Machine Learning Reconstruction of Neutrino-LAr Interactions

Sequential or chained models are increasingly prevalent in machine learning for scientific applications, due to their flexibility and ease of development. Chained models are particularly useful when a task is separable into distinct steps with a hierarchy of meaningful intermediate representations. In reliability-critical tasks, it is important to quantify the confidence of model inferences. However, chained models pose an additional challenge for uncertainty quantification, especially when input uncertainties need to be propagated. In such cases, a fully uncertainty-aware chain of models is required, where each step accepts a probability distribution over the input space, and produces a probability distribution over the output space. In this work, we present a case study for adapting a single model within an existing chain, designed for reconstruction within neutrino-Argon interactions, developed for neutrino oscillation experiments such as MicroBooNE, ICARUS, and the future DUNE experiment. We test the performance of an input uncertainty-enabled model against an uncertainty-blinded model using a method for generating synthetic noise. By comparing these two, we assess the increase in inference quality achieved by exposing models to upstream uncertainty estimates.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1975

Distribution and moments of radial error

An investigation of the moments and probability distribution of the resultant of two normally distributed random variables is presented. This is the so-called generalized Rayleigh distribution which has many applications in the study of wind shear, random noise, and radar. The most general formula was derived, and two special cases were considered for which tables of the moments and probability distribution functions are included as an appendix. One of the special cases was generalized to n-dimensions.

White, R. G.↗

NASA NTRS1988

The topology of fractal universes

It is shown how the genus per unit volume of isodensity surfaces in general nonlinear universes is related to the entire hierarchy of correlation functions. The general relation between the correlation function, the probability distribution of densities at several points, and the probability distributions of density and its derivatives at a point are given. Formulas for the area and genus per unit volume of isodensity surfaces are presented. As an application, after first reviewing the case of Gaussian fields, analytic results are reported for one particular example of a thoroughly nonlinear universe, Mandelbrot's Rayleigh-Levy random-walk fractal. While this fractal bears little resemblance to the real universe of galaxies, it possesses the singular and theoretically interesting property that in it cluster-cluster correlations are identically equal to galaxy-galaxy correlations to all orders.

Hamilton, A. J. S.↗

NASA NTRS1972

The prediction of millimeter wavelength precipitation attenuation using a cylindrical storm cell model

The single terminal and two terminal diversity precipitation attenuation distributions are calculated for a millimeter wavelength earth-satellite propagation path. The calculated probability distributions are compared to attenuation probability distributions measured at two locations with the ATS-5 millimeter wavelength experimental 15.63 GHz downlink. Using an empirically derived three mode exponential cell diameter-rain rate function, the calculated distributions agreed within -2 to +1 db of the measured distributions for both terminal locations. The sensitivity of the calculated distribution to variations of the cell model parameters was also investigated.

Zintsmaster, L. R.↗

NASA NTRS2003

A Rational Approach to Determine Minimum Strength Thresholds in Novel Structural Materials

Design of safe and survivable structures requires the availability of guaranteed minimum strength thresholds for structural materials to enable a meaningful comparison of strength requirement and available strength. This paper develops a procedure for determining such a threshold with a desired degree of confidence, for structural materials with none or minimal industrial experience. The problem arose in attempting to use a new, highly weight-efficient structural load tendon material to achieve a lightweight super-pressure balloon. The developed procedure applies to lineal (one dimensional) structural elements. One important aspect of the formulation is that it extrapolates to expected probability distributions for long length specimen samples from some hypothesized probability distribution that has been obtained from a shorter length specimen sample. The use of the developed procedure is illustrated using both real and simulated data.

Schur, Willi W.↗

NASA NTRS2004

Derivation of Failure Rates and Probability of Failures for the International Space Station Probabilistic Risk Assessment Study

National Aeronautics and Space Administration s (NASA) International Space Station (ISS) Program uses Probabilistic Risk Assessment (PRA) as part of its Continuous Risk Management Process. It is used as a decision and management support tool to not only quantify risk for specific conditions, but more importantly comparing different operational and management options to determine the lowest risk option and provide rationale for management decisions. This paper presents the derivation of the probability distributions used to quantify the failure rates and the probability of failures of the basic events employed in the PRA model of the ISS. The paper will show how a Bayesian approach was used with different sources of data including the actual ISS on orbit failures to enhance the confidence in results of the PRA. As time progresses and more meaningful data is gathered from on orbit failures, an increasingly accurate failure rate probability distribution for the basic events of the ISS PRA model can be obtained. The ISS PRA has been developed by mapping the ISS critical systems such as propulsion, thermal control, or power generation into event sequences diagrams and fault trees. The lowest level of indenture of the fault trees was the orbital replacement units (ORU). The ORU level was chosen consistently with the level of statistically meaningful data that could be obtained from the aerospace industry and from the experts in the field. For example, data was gathered for the solenoid valves present in the propulsion system of the ISS. However valves themselves are composed of parts and the individual failure of these parts was not accounted for in the PRA model. In other words the failure of a spring within a valve was considered a failure of the valve itself.

Vitali, Roberto↗

NASA NTRS2020

Robustness Computation of Dynamic Controllability in Probabilistic Temporal Networks with Ordinary Distributions

Most existing works in Probabilistic Simple Temporal Networks (PSTNs) base their frameworks on well-defined probability distributions. This paper addresses on PSTN Dynamic Controllability (DC) robustness measure, i.e. the execution success probability of a network under dynamic control.We consider PSTNs where the probability distributions of the contingent edges are ordinary distributed (e.g. non-parametric, non-symmetric). We introduce the concepts of dispatching protocol (DP) as well as DP-robustness, the probability of success under a predefined dynamic policy.We propose a fixed-parameter pseudo-polynomial time algorithm to compute the exact DP-robustness of any PSTN under \textit{NextFirst} protocol, and apply to various PSTN datasets, including the real case of planetary exploration in the context of the Mars 2020 rover, and propose an original structural analysis.

Saint-Guillain, Michael↗

NASA NTRS1993

Bayesian image reconstruction - The pixon and optimal image modeling

In this paper we describe the optimal image model, maximum residual likelihood method (OptMRL) for image reconstruction. OptMRL is a Bayesian image reconstruction technique for removing point-spread function blurring. OptMRL uses both a goodness-of-fit criterion (GOF) and an 'image prior', i.e., a function which quantifies the a priori probability of the image. Unlike standard maximum entropy methods, which typically reconstruct the image on the data pixel grid, OptMRL varies the image model in order to find the optimal functional basis with which to represent the image. We show how an optimal basis for image representation can be selected and in doing so, develop the concept of the 'pixon' which is a generalized image cell from which this basis is constructed. By allowing both the image and the image representation to be variable, the OptMRL method greatly increases the volume of solution space over which the image is optimized. Hence the likelihood of the final reconstructed image is greatly increased. For the goodness-of-fit criterion, OptMRL uses the maximum residual likelihood probability distribution introduced previously by Pina and Puetter (1992). This GOF probability distribution, which is based on the spatial autocorrelation of the residuals, has the advantage that it ensures spatially uncorrelated image reconstruction residuals.

Pina, R. K.↗

NASA NTRS2008

Simulation of Stochastic Processes by Coupled ODE-PDE

A document discusses the emergence of randomness in solutions of coupled, fully deterministic ODE-PDE (ordinary differential equations-partial differential equations) due to failure of the Lipschitz condition as a new phenomenon. It is possible to exploit the special properties of ordinary differential equations (represented by an arbitrarily chosen, dynamical system) coupled with the corresponding Liouville equations (used to describe the evolution of initial uncertainties in terms of joint probability distribution) in order to simulate stochastic processes with the proscribed probability distributions. The important advantage of the proposed approach is that the simulation does not require a random-number generator.

Zak, Michail↗

NASA NTRS2008

A New Bond Albedo for Performing Orbital Debris Brightness to Size Transformations

We have developed a technique for estimating the intrinsic size distribution of orbital debris objects via optical measurements alone. The process is predicated on the empirically observed power-law size distribution of debris (as indicated by radar RCS measurements) and the log-normal probability distribution of optical albedos as ascertained from phase (Lambertian) and range-corrected telescopic brightness measurements. Since the observed distribution of optical brightness is the product integral of the size distribution of the parent [debris] population with the albedo probability distribution, it is a straightforward matter to transform a given distribution of optical brightness back to a size distribution by the appropriate choice of a single albedo value. This is true because the integration of a powerlaw with a log-normal distribution (Fredholm Integral of the First Kind) yields a Gaussian-blurred power-law distribution with identical power-law exponent. Application of a single albedo to this distribution recovers a simple power-law [in size] which is linearly offset from the original distribution by a constant whose value depends on the choice of the albedo. Significantly, there exists a unique Bond albedo which, when applied to an observed brightness distribution, yields zero offset and therefore recovers the original size distribution. For physically realistic powerlaws of negative slope, the proper choice of albedo recovers the parent size distribution by compensating for the observational bias caused by the large number of small objects that appear anomalously large (bright) - and thereby skew the small population upward by rising above the detection threshold - and the lower number of large objects that appear anomalously small (dim). Based on this comprehensive analysis, a value of 0.13 should be applied to all orbital debris albedo-based brightness-to-size transformations regardless of data source. Its prima fascia genesis, derived and constructed from the current RCS to size conversion methodology (SiBAM Size-Based Estimation Model) and optical data reduction standards, assures consistency in application with the prior canonical value of 0.1. Herein we present the empirical and mathematical arguments for this approach and by example apply it to a comprehensive set of photometric data acquired via NASA's Liquid Mirror Telescopes during the 2000-2001 observing season.

Mulrooney, Mark K.↗

NASA NTRS1972

Correlation signatures of wet soils and snows

Interpretation, analysis, and development of algorithms have provided the necessary computational programming tools for soil data processing, data handling and analysis. Algorithms that have been developed thus far, are adequate and have been proven successful for several preliminary and fundamental applications such as software interfacing capabilities, probability distributions, grey level print plotting, contour plotting, isometric data displays, joint probability distributions, boundary mapping, channel registration and ground scene classification. A description of an Earth Resources Flight Data Processor, (ERFDP), which handles and processes earth resources data under a users control is provided.

Phillips, M. R.↗

NASA NTRS1999

Disturbance Dynamics in Transitional and Turbulent Flows

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. First the decay of laminar disturbances and turbulence in mean shear-free flows is studied. In laminar flows, such disturbances are linear superpositions of modes governed by the Orr-Sommerfeld equation. In turbulent flows, disturbances are described through transport equations for representative mean quantities. The link between a description based on a deterministic evolution equation and a probability based mean transport equation is established. Because an uncertainty in initial conditions exists in the laminar as well as the turbulent regime, a probability distribution must be defined even in the laminar case. Using this probability distribution, it is shown that the exponential decay of the linear modes in the laminar regime can be related to a power law decay of both the (ensemble) mean disturbance kinetic energy and the dissipation rate. The evolution of these mean disturbance quantities is then described by transport equations similar to those for the corresponding turbulent decaying flow. Second, homogeneous shear flow, where disturbances can be described by rapid distortion theory (RDT), is studied. The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined.

Grosch, Chester E.↗

NASA NTRS1994

Velocity distributions among colliding asteroids

The probability distribution for impact velocities between two given asteroids is wide, non-Gaussian, and often contains spikes according to our new method of analysis in which each possible orbital geometry for collision is weighted according to its probability. An average value would give a good representation only if the distribution were smooth and narrow. Therefore, the complete velocity distribution we obtain for various asteroid populations differs significantly from published histograms of average velocities. For all pairs among the 682 asteroids in the main-belt with D greater than 50 km, we find that our computed velocity distribution is much wider than previously computed histograms of average velocities. In this case, the most probable impact velocity is approximately 4.4 km/sec, compared with the mean impact velocity of 5.3 km/sec. For cases of a single asteroid (e.g., Gaspra or Ida) relative to an impacting population, the distribution we find yields lower velocities than previously reported by others. The width of these velocity distributions implies that mean impact velocities must be used with caution when calculating asteroid collisional lifetimes or crater-size distributions. Since the most probable impact velocities are lower than the mean, disruption events may occur less frequently than previously estimated. However, this disruption rate may be balanced somewhat by an apparent increase in the frequency of high-velocity impacts between asteroids. These results have implications for issues such as asteroidal disruption rates, the amount/type of impact ejecta available for meteoritical delivery to the Earth, and the geology and evolution of specific asteroids like Gaspra.

Bottke, William F., Jr.↗

NASA NTRS2000

A simulator for evaluating methods for the detection of lesion-deficit associations

Although much has been learned about the functional organization of the human brain through lesion-deficit analysis, the variety of statistical and image-processing methods developed for this purpose precludes a closed-form analysis of the statistical power of these systems. Therefore, we developed a lesion-deficit simulator (LDS), which generates artificial subjects, each of which consists of a set of functional deficits, and a brain image with lesions; the deficits and lesions conform to predefined distributions. We used probability distributions to model the number, sizes, and spatial distribution of lesions, to model the structure-function associations, and to model registration error. We used the LDS to evaluate, as examples, the effects of the complexities and strengths of lesion-deficit associations, and of registration error, on the power of lesion-deficit analysis. We measured the numbers of recovered associations from these simulated data, as a function of the number of subjects analyzed, the strengths and number of associations in the statistical model, the number of structures associated with a particular function, and the prior probabilities of structures being abnormal. The number of subjects required to recover the simulated lesion-deficit associations was found to have an inverse relationship to the strength of associations, and to the smallest probability in the structure-function model. The number of structures associated with a particular function (i.e., the complexity of associations) had a much greater effect on the performance of the analysis method than did the total number of associations. We also found that registration error of 5 mm or less reduces the number of associations discovered by approximately 13% compared to perfect registration. The LDS provides a flexible framework for evaluating many aspects of lesion-deficit analysis.

NASA Program Biomedical Research and Countermeasur↗